Question

Read the story and respond to the requests below. Use the Format Editor x−−√ to type math or use the Image icon to upload a photo of your work.

You begin doing 20 sit-ups every day in November. You do 32 sit-ups every day in December, 44 sit-ups every day in January, and so on.

Write the sequence represented by the story.
Is the sequence arithmetic or geometric?
Does it have a common difference or common ratio?
What is the common difference or common ratio?
What are the next three terms?

Answer 1

Let's analyze the story step by step.

### Sequence Representation:
The sequence starts with:
- November: 20 sit-ups
- December: 32 sit-ups
- January: 44 sit-ups

Following this pattern, we can see that the number of sit-ups increases each month.

The terms of the sequence are:
- First term \( a_1 = 20 \)
- Second term \( a_2 = 32 \)
- Third term \( a_3 = 44 \)

To write out the terms more clearly, we note that:
- \( a_4 = 56 \) (following the established pattern)
- \( a_5 = 68 \)
- \( a_6 = 80 \)

### Sequence Formula:
Now we can represent the sequence as:
\[ 20, 32, 44, 56, 68, 80, \ldots \]

### Type of Sequence:
To determine if this sequence is arithmetic or geometric, we'll look for a consistent additive or multiplicative pattern.

1. **Arithmetic Sequence:** If the difference between consecutive terms is constant.
2. **Geometric Sequence:** If the ratio between consecutive terms is constant.

**Analysis:**
- The difference from 20 to 32 is \( 32 - 20 = 12 \).
- The difference from 32 to 44 is \( 44 - 32 = 12 \).
- The difference from 44 to 56 is \( 56 - 44 = 12 \).

Since the differences between consecutive terms are constant, this sequence is **arithmetic**.

### Common Difference or Ratio:
- **Common Difference:** The difference between consecutive terms.
- **Common Ratio:** The ratio of consecutive terms (used only in geometric sequences).

**In this case:**
- **Common Difference:** \( d = 12 \)

### Finding the Next Three Terms:
Now we can find the next three terms in the sequence:
- From 80, the next term will be \( 80 + 12 = 92 \).
- The term after that will be \( 92 + 12 = 104 \).
- The third subsequent term will be \( 104 + 12 = 116 \).

### Summary of Responses:
1. **Sequence:** \( 20, 32, 44, 56, 68, 80, \ldots \)
2. **Type:** Arithmetic
3. **Common Difference:** \( 12 \)
4. **Next Three Terms:** \( 92, 104, 116 \)